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In this work, we investigate diverse analytical solutions and modulation instability of the nonlinear Schrödinger equation with an anti-cubic nonlinear term. We use the traveling wave transformation and the New Generalized Extended Direct Algebraic Method to perform a variety of exact traveling wave soliton-like solutions. The obtained solutions are among other trigonometric function solutions, complex soliton solutions, and rational solutions. A particular behavior has been depicted in self-focusing and defocusing nonlinearity where dark soliton changes to super rogue waves and breather soliton is obtained. We then use the linearizing scheme to get the growth rate of modulation instability. For a specific value of the excitation wave number and a specific time of simulation, diverse waveforms of the modulated waves are obtained. We have demonstrated that the combined dispersion and nonlinear terms, as well as the anti-cubic nonlinear term, can develop modulation instability. To confirm the predictions based on the analytical results, we use the direct numerical simulation, from which we have displayed a modulation wave of the growth rate modulation instability. It results equally that the anti-cubic nonlinearity is an important object to control the propagation of the solitonic waves in the nonlinear systems. [ABSTRACT FROM AUTHOR]

This work examines the effects of the higher-order dispersion (HOD) on modulated waves and modulation instability (MI) gain by using the perturbed nonlinear Schrödinger equation having the cubic quintic septic nonlinearity. We use the new generalized auxiliary equation method to underline the behavior of bright, dark soliton as well as the combined optical soliton solutions. Considering the constraint relation on the combined optical soliton solutions, we carry out the W-shaped profile. We observe that the obtained optical soliton solutions can spread without alteration and can preserve their shapes. Moreover, we enhance the obtained analytical results by using the Split-Step Fourier technic which yields points out stable optical solitons. We have obviously highlighted the effects of the HOD by means of the numerical simulation. Owing to the fact that the model has nonlinear and dispersions terms, we have shown how the MI growth rate and the modulation bands can expand under the influence of the HOD terms in normal and anomalous dispersion regimes. It is worth mentioning the appearance of the unstable zones when the value of the higher-order dispersion term increases and additional bands also emerges. Compared these results with some previous works (Kohl in Optik 203:163451, 2020; Yao in Res Phys 30:104825, 2021; Wang in Phys Lett A 372:417–423, 2008; Ismail in Appl Math Comput 209:425–429, 2009; Nestor in Eur Phys J Plus 135:380, 2020), additional bands of the MI have been pointed out while the MI gain spectra have shown new sides lobes. We expect that these relevant and concise results could probably help to improve the communication system through optic fibers. [ABSTRACT FROM AUTHOR]

Investigation of soliton solutions and Modulation Instability (MI) analysis to the (2+1)-Stochastic Chiral Nonlinear Schrödinger Equation (SCNLSE) having noise force and Brownian movement (BM) have been done in this work. We first use the transformation technic and auxiliary equation method (AEM) to establish three types of soliton solutions. Depending on the constraint relation set on the auxiliary equation parameters, it is obtained bright and dark soliton solutions. Thereafter, taking the same different parameters it is gained the combined bright-dark soliton, bright-bright soliton solutions. By employing suitable values of the used parameters together with the model parameters it is show out W-shaped soliton solutions. Throughout the paper it is clearly established the impact of the noise strength and BM on the obtained W-shaped soliton. To determine the stability or instability zones, it was used the linear stability tool. It is shown that by varying the different parameters the Modulation bands can grow instability and stability zones can also emerge. These results permit to consolidate the previous works (Albosaily et al. in symmetry 12:1874, 2020, Abdelrahman et al. in AIMS Math 6:2970, 2021) exposed on the effects of noise on the soliton and the MI gain is new in this context. [ABSTRACT FROM AUTHOR]

In this paper, we propose a recursive Darboux transformation in a generalized form of a focusing vector Nonlinear Schrödinger Equation (NLSE) known as the Manakov System. We apply this generalized recursive Darboux transformation to the Lax-pairs of this system in view of generating the Nth-order vector generalization rogue wave solutions with a rule of iteration. We discuss from first- to three-order vector generalizations of rogue wave solutions while illustrating these features with some 3D, 2D graphical depictions. We illustrate a clear connection between higher-order rogue wave solutions and their free parameters for better understanding the physical phenomena described by the Manakov system [ABSTRACT FROM AUTHOR]

In this paper, modulation instability, bifurcation analysis, and soliton solutions are investigated in nonlinear media with odd-order dispersion terms. A generalized nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity is considered. A linear analysis method is used to derive an expression of the modulation instability spectrum, and the effects of the fourth-order and group velocity dispersion are pointed out on the modulation instability bands. The results show that for negative values of the fourth order in a normal dispersion regime, the modulation instability vanishes. Another important aspect is revealed when the fourth-order dispersion gets positive and has high values; additional bands emerge to enlarge the bandwidths of the modulation instability. To further confirm the effects of the odd-order dispersion in the nonlinear structure, the bifurcation, phase portraits, and chaotic behaviors have been investigated to show how instability arises in the nonlinear structure. Using numerical simulation, in particular the Runge-Kutta algorithm, the sensitivity of nonlinear systems is pointed out to confirm an unstable behavior. This investigation confirms once again the fact that the modulation spectrum is sensitive to higher-order dispersion in normal and anomalous dispersion regimes. A traveling wave hypothesis is employed to lead to the direct integration of the nonlinear system, and some specific soliton solutions are extracted. For particular constraint conditions on the discriminant, bright and dark solitons as well as Jacobi elliptic function solutions emerge. The obtained results could be used to improve the transmission signal via the optical fiber and secure the data in communication systems. • Modulation instability, bifurcation analysis, and soliton solutions are investigated in this paper. • A generalized nonlinear Schrodinger equation with quartic dispersion and cubic-quintic nonlinearity is considered. • The effects of the fourth-order and group velocity dispersion are pointed out on the modulation instability bands. [ABSTRACT FROM AUTHOR]

In this paper, we examined the effects of the nonlinear parameters and driven amplitude on the training envelope in discrete cubic-quintic nonlinear Schrödinger equation with arbitrarily high-order nonlinearities. From the numerical simulation, we exhibited the generation of the train of pulses and dark soliton when the driven amplitude is above the threshold. For a specific time of propagation, we illustrate how the variation of the driven amplitude and the nonlinear cubic-quintic parameters can generate instability in the forbidden gap. It emerges that the model of the discrete nonlinear Schrödinger equation with arbitrarily high-order nonlinearities can be used to generate the train of waves despite the fact that it is not integrable in the continuum limit approximation. We also displayed the effects of the cubic-quintic terms on the modulation instability growth rate. The obtained results will open new features to the train of pulses and dark soliton in the nonlinear fibers optics. • We studied discrete cubic-quintic nonlinear Schrödinger equation with arbitrarily high-order nonlinearities. • We examined the effects of the nonlinear parameters and driven amplitude on the training envelope in this nonlinear model. • We illustrated the effects of the cubic-quintic terms on the modulation instability growth rate. • The obtained solutions are depicted by 2D and 3D plots. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the modulated wave and W-shaped profile in birefringent fibers having the anti-cubic nonlinearity terms. We use the traveling wave hypothesis to show out the velocity of the soliton and the constraint relation on the anti-cubic nonlinear terms. We use the Jacobi elliptic function solutions to point out two types of combined solutions. After some assumption on the modulus of the Jacobi elliptic function, we have shown out the combined bright-bright soliton and dark-dark soliton-like solutions. We use the linearizing algorithm to analyze the modulation instability (MI) growth rate. We have shown that the anti-cubic nonlinear terms and cross-phase modulation (XPM) can increase MI bands and the amplitude of the MI growth rate. To corroborate the prediction made on analytical results, we use the numerical investigation to show the propagation of the modulated wave and W-shaped profile in terms of cell index. We exhibited through the numerical results that the modulated wave can conserve high energy during its propagation in birefringent fibers. The obtained results will certainly open new perspectives in optical fibers during the transmission of huge data. • The modulated wave and W-shaped profile in birefringent fibers having the anti-cubic nonlinearity terms is studied. • The linear stability analysis is used. • The derived solutions are displayed using 2D and 3D plots. [ABSTRACT FROM AUTHOR]

• The nonlinear discrete equation is studied. • The rational w- shape solitons are retrieved. • The modulation analysis is presented. • The obtained solutions are presented by 2D and 3D figures. This paper examines envelope solitons, rational W-shape solitons to a Nonlinear Electrical Transmission Line (NETL) with Josephson junction (JJ) that consist of the N t h cells of circuits. By employing the reductive perturbation approach in the semi-discrete approximation, we obtain the nonlinear Schrödinger equation (NLSE). From this equation, the frequency ranges of propagation of bright and dark solitons were obtained. As in most of the works, the NLSE obtained is well known as the seat of rogue waves and Peregrine solitons. The nonlinearity provided by the JJ, combined with the dispersion, made it possible to study the Modulation Instability (MI) gain spectrum. The 3D and 2D graphical representations illustrated the instability zones and the W-rational solutions have followed by using numerical simulation. The results obtained in this paper are of a very capital contribution for the study of rogue waves in nonlinear transmission lines or other physical problems. [ABSTRACT FROM AUTHOR]

• The Coupled Nonlinear Schrödinger Equation (CNLSE) throughout Kerr law nonlinearity is considered. • The linear stability analysis is used. • The modulation instability analysis is presented. • The obtained results are showed by 2D and 3D Figures. We investigate diverse optical solitons and other solution for the Coupled Nonlinear Schrödinger Equation (CNLSE) having Four-Wave Mixing (FWM) and Kerr nonlinearity term. We stress the effects of different parameters of the model on the obtained optical soliton by employing the New Generalized Extended Direct Algebraic Method (NGEDAM). The solutions contain W-shaped profile of the bright optical soliton, dark optical soliton and diverse other solutions including the combined complex optical soliton. Interestingly, W-shaped optical soliton and dark optical soliton own a good argument for long communication and big data transmission originating from the self-steepening, Kerr nonlinearity effects and FWM. We also, emphasize the Modulation Instability (MI) gain spectra to narrow the impact of the FWM in birefringence. Thereafter, we involve the ellipticity angle effects in normal and anomalous dispersion regime on the MI gain spectra. Parametric conditions for the existence of the obtained optical soliton solutions and MI gain are given. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the propagation of the modulated waves (MWs) patterns with the effects of the external field (EF) in a photorefractive crystal. To do so, we use the coupled nonlinear Schrödinger equations and we apply the auxiliary equation method to point out four families of solutions such as bright, dark, combined bright-dark, and bright-bright soliton. We pointed out the effects of EF on the modulation instability growth rate. Throughout the numerical simulation, we show that EF can generate MWs patterns when the short and open circuit condition is considered in the case of bright soliton. However, when we consider dark soliton solution as an initial condition and we have only obtained the MW pattern in the case of a short circuit. It results in this study of the propagation of the MWs patterns and chaos-like motion for specific cell indexes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the modulation instability and the dynamics of solitary waves in a higher-order nonlinear Schrödinger equation. We use a powerful and robust method known as the modified sub-equation method to secure the solitary waves and other numerous solutions to this nonlinear model. Several constraint conditions that guarantee the existence of these solutions are highlighted. Utilizing a linearizing technique, we establish the modulation instability gain. The influence of the higher nonlinear component on the modulation instability is also discussed. Furthermore, we perform the numerical simulations of this model by using the split-step Fourier method. Finally, several solutions are shown in two and three dimensions to better explain the behavior of the considered model. The two-dimensional plots depict Peregrine soliton under the influence of the nonlinear dispersion term. These graphs are useful for understanding the dynamic properties of the obtained results. • New optical solitons for the HNSE were constructed. • The obtained solutions were found using the modified sub-equation technique. • The model's modulation instability is studied. • The physical meaning for the solutions is graphically investigated. [ABSTRACT FROM AUTHOR]